I knew about a white-glove inspection, but a fingertip search was new to me.

Web examples suggest that it's an expression used by the British police, e.g. in this BBC story from August 19, 2002, "Fingertip search for clues", which describes such a search of a scene where bodies were found:

In investigations of this kind, scenes of crime officers cordon off the area and decide what type of experts are needed to help in the operation.

The atmosphere in the copse would be both quiet and efficient, according to Professor Anthony Busuttil, of Edinburgh University's department of pathology.

With 33 years experience in this highly specialised area, he knows better than most the difficulties involved in such cases.

"It's very hard work, emotionally, physically and mentally draining but someone must do it," he said.
Forensic investigations are strictly timetabled and co-ordinated.

"There is a lot of activity at the scene, but it's very quiet because everyone needs to concentrate extremely hard." said Professor Busuttil.

Wearing goggles, gloves and disposable sterile paper or plastic suits, officers crawl towards the bodies from a radius of several metres.

They pick up everything found with the naked eye while a police photographer and cameraman take pictures and video to help psychologists build a character profile of the killer.

An exhibits officer separately collects and places articles in sealed and labelled polythene bags.
Once the fingertip search is complete, raised aluminium or wooden platforms are erected to reach the bodies without disturbing the earth beneath.

So the fingertips are the least of it, it seems.

In the first few pages of
returns from Google (where my favorite is the
job opening in the West Midlands where "you will oversee the installation of amphibian fencing, undertake a fingertip search for great crested newts, check traps, oraganise bat surveys and ensure that there is no negative effect to the badger population"), I don't find any examples of the phrase "fingertip search" being used by American sources. I'm sure that American law enforcement teams carry out searches of the same kind -- I wonder what they call them.